P ARAMETRIC E QUATIONS Section 10-2

Plane Curves and Parametric Equations Consider the path of an object that is propelled into air at an angle of 45°. The object travels a parabolic path given by This tells us the where the object has been, it doesn’t tell us when the object was at a given point (x,y)

Plane Curves and Parametric Equations To determine the time we use a third variable t, called a parameter By writing both x and y as functions of t, we obtain the parametric equations Together, the parametric equations and the graph are called the plane curve.

1) – Sketching a Curve Sketch the curve described by the parametric equations Solution: For values of t on the given interval, the parametric equations yield the points (x, y) shown in the table.

Example 1 – Solution By plotting these points in order of increasing t and using the continuity of f and g, we obtain the curve C. *Note that the arrows on the curve indicate its orientation as t increases from –2 to 3. cont’d

Finding a rectangular equation that represents the graph of a set of parametric equations is called eliminating the parameter. Eliminating the Parameter Once you have eliminated the parameter, you can recognize that the equation x = 4y 2 – 4 represents a parabola with a horizontal axis and vertex at (–4, 0)

2) Find the Cartesian rectangular equation for the parametric equations

3) Find the rectangular equation by eliminating the parameter Find the domain and Range txy

4) Eliminate the parameter and find the corresponding rectangular equation for

4)cont. Graph the curve

H OME W ORK Page 718 # 3-15 odd, 21, 25, 29, and 68